INP pattern
Fix some theory T . Let \kappa be a cardinal. An inp pattern of depth \kappa is a collection of formulas \langle \phi_\alpha(x;y) \rangle_{\alpha < \kappa} and constants b_{\alpha,i} for \alpha < \kappa and i < \omega and integers k_\alpha < \omega such that for every \alpha < \kappa , the set of formulas \{\phi(x;b_{\alpha,i}) : i < \omega\} is k_\alpha -inconsistent, but for every function \eta : \kappa \to \omega , the collection \{\phi(x;b_{\alpha,\eta(\alpha)}) : \alpha < \kappa\} is consistent. More generally, if \Sigma(x) is a partial type, an inp pattern of depth \kappa in \Sigma(x) is an inp pattern of depth \kappa such that for every \eta : \kappa \to \omega , \Sigma(x) \cup \{\phi(x;b_{\alpha,\eta(\alpha)}) : \alpha < \kappa\} is consistent. Shelah defines \kappa_{inp} of a theory to be the supremum of the depths of possible inp-patterns. Hans Adler (right?) defines the burden of a partial type \Sigma(x) to be the supremum of the depths of the inp patterns in \Sigma(x) . A theory is said to be strong if there are no inp patterns of depth \omega . A theory is NTP_2 if and only if \kappa_{inp} < \infty . Artem Chernikov (right?) proved that burden is submultiplicative in the following sense: if bdn(b/C) < \kappa and bdn(a/bC) < \lambda , then bdn(ab/C) < \kappa \times \lambda . It is conjectured that burden is subadditive ( bdn(ab/C) \le bdn(a/bC) + bdn(b/C) ), but this is unknown. Given an inp pattern of depth \kappa , one can always find an inp pattern of the same depth, using the same formulas and same k_\alpha 's, such that the rows \langle b_{\alpha,i}\rangle_{i < \omega} are mutually indiscernible. Given mutual indiscernibility, the k_\alpha -inconsistence can be rephrased as inconsistency. And the only vertical path one must check is the leftmost column. So one may also define the burden of \Sigma(x) to be the supremum of the \kappa for which there exists \kappa mutually indiscernible sequences \langle b_{\alpha,i} \rangle_i for \alpha < \kappa and formulas \phi_\alpha(x;y) for \alpha < \kappa such that for each \alpha , \{\phi_\alpha(x;b_{\alpha,i}) : i < \omega\} is inconsistent, and \Sigma(x) \cup \{\phi_\alpha(x;b_{\alpha,0}) : \alpha < \kappa\} is consistent. Relation to ict patterns Any mutually indiscernible inp pattern is already a mutually indiscernible ict pattern. Under the hypothesis of NIP, a mutually indiscernible ict pattern of depth \kappa can be converted to a mutually indiscernible inp pattern of the same depth, as follows. If the original ict pattern is \{\phi_\alpha(x;b_{\alpha,i}) , then we take as our inp pattern the array of formulas whose entry in the \alpha th row and i th column is \phi_\alpha(x;b_{\alpha,2i}) \wedge \neg \phi_\alpha(x;b_{\alpha,2i+1}) . The "no alternation" characterization of NIP implies that each row is inconsistent. The ict condition ensures that we can find an a satisfying \phi_\alpha(x;b_{\alpha,0}) and \neg \phi_\alpha(x;b_{\alpha,1}) for every \alpha , showing that the first column is consistent. Consequently, if NIP holds (equivalently, \kappa_{ict} < \infty ), then \kappa_{inp} = \kappa_{ict} , and the burden of any type equals its dp-rank. Also, a theory is strongly dependent if and only if it is strong and NIP (dependent).